Cospectral cluster quiver mutation conjecture

Let $Q$ be a cluster quiver, and let $\operatorname{Mut}(Q)$ denote its mutation class. Two quivers are cospectral when they have the same exchange spectrum; quivers are isomorphic when they are isomorphic as cluster quivers. A mutation at a sink or source is a mutation at a vertex that is respectively a sink or a source.

Cospectral cluster quiver mutation conjecture. The quivers $Q',Q”\in\operatorname{Mut}(Q)$ are cospectral if and only if there exists a cluster quiver $R\in\operatorname{Mut}(Q)$ such that $Q'$ and $R$ are isomorphic and $Q”$ can be obtained from $R$ by mutation at sinks and sources.

This is the formal version of the preceding informal statement about finite connected families of cospectral cluster quivers. The source supplies no resolution status or further evidence.